Density Estimation for Censored Data.

Abstract

In scientific research, experimenters frequently encounter censored data. Censoring is especially a concern in clinical trials. In such structures, some patients may withdraw from the study, others may die from competing risks and the study may end before all results are known for the remaining patients. We are interested in the censored data due to competing risk. Kaplan-Meier (1958) proposed the product limit estimate. They partitioned the interval {0,T} into sequences of intervals and estimate the probability of surviving in each interval separately. As a result, the probability of surviving beyond time T is estimated by the product of these probabilities. One assumption, which frequently is inherent in the problem, is the existence of probability density with respect to lebesgue measure. In this case, estimation of the density is an obvious problem. In Chapter One and Two, we use histogram and product limit estimate to estimate the underlying density and make suggestion for choosing the interval width. If the sample size, n, is large, the optimal width is of order n('-1/3) and the integrated mean square error is of order n('-2/3). In Chapter Three, we impose stronger conditions on the underlying distributions and we estimate the density by frequency polygon. The optimal interval width and integrated mean square error are of order n('-1/5) and n('-4/5) respectively. In many cases, the survival times are related to some covariates. We investigate the non-linear regression between the survival time and covariates in Chapter Four. We make recommendation to choose the optimal interval width. They are found to be of order n('-1/3) and the corresponding error is found to be of order n('-2/3). Computer simulations are in Chapter Five and the results are satisfactory.